Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
575 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#576 | C2 | semidirect |
| 2. C2×#509 | C2 | semidirect |
| 3. C22×D16 | C2 | semidirect |
| 4. C2×#472 | C2 | semidirect |
| 5. #2192 | C2 | direct (N × H) |
| 6. C2×#361 | C2 | semidirect |
| 7. C2×#359 | C2 | semidirect |
| 8. C2×#350 | C2 | semidirect |
| 13. #576 | C22 | semidirect |
| 21. C22×D8 | C22 | semidirect |
| 24. #509 | C22 | semidirect |
| 29. C2×D16 | C22 | semidirect |
| 30. C2×M32 | C22 | semidirect |
| 31. C22×C16 | C22 | semidirect |
| 33. #472 | C22 | semidirect |
| 34. C2×#108 | C22 | semidirect |
| 36. #361 | C22 | semidirect |
| 37. #359 | C22 | semidirect |
| 38. #350 | C22 | semidirect |
| 84. C2×D8 | C23 | semidirect |
| 98. M32 | C23 | semidirect |
| 102. C2×C16 | C23 | semidirect |
| 104. #108 | C23 | semidirect |
| 270. C2 | #2192 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#576 | 2 |
| C2×#509 | 2 |
| C22×D16 | 2 |
| C2×#472 | 2 |
| #2192 | 2 |
| C2×#361 | 2 |
| C2×#359 | 2 |
| C2×#350 | 2 |